When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators under mild conditions on the data collection process and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.

翻译：当数据以自适应方式收集时，即使是普通最小二乘法等简单方法也可能表现出非正态渐近行为。这一不良后果导致基于渐近正态性的假设检验和置信区间可能得出错误结论。我们提出了一类在线去偏估计器族，以纠正最小二乘估计中的这些分布异常现象。所提方法利用数据集中存在的协方差结构，在信息积累更充分的方向上提供更精确的估计。我们证明在数据收集过程的温和条件下，所提在线去偏估计器具有渐近正态性，并给出渐近精确的置信区间。此外，我们证明了自适应线性回归问题的极小极大下界，为估计器比较建立了基准。在所提估计器满足多种条件的情况下，其可达极小极大下界。通过应用于多臂老虎机、自回归时间序列估计以及含探索的主动学习，我们验证了理论的有效性。